Application of metaheuristic algorithms has been of continued interest in the field of electrical engineering because of their powerful features. In this work special design is done for a tapered transmission line used for matching an arbitrary real load to a 50{\Omega} line. The problem at hand is to match this arbitrary load to 50 {\Omega} line using three section tapered transmission line with impedances in decreasing order from the load. So the problem becomes optimizing an equation with three unknowns with various conditions. The optimized values are obtained using Particle Swarm Optimization. It can easily be shown that PSO is very strong in solving this kind of multiobjective optimization problems.
Multiobjective design optimization problems require multiobjective optimization techniques to solve, and it is often very challenging to obtain high-quality Pareto fronts accurately. In this paper, the recently developed flower pollination algorithm (FPA) is extended to solve multiobjective optimization problems. The proposed method is used to solve a set of multobjective test functions and two bi-objective design benchmarks, and a comparison of the proposed algorithm with other algorithms has been made, which shows that FPA is efficient with a good convergence rate. Finally, the importance for further parametric studies and theoretical analysis are highlighted and discussed.
Recently, there has been a renewed interest in decomposition-based approaches for evolutionary multiobjective optimization. However, the impact of the choice of the underlying scalarizing function(s) is still far from being well understood. In this paper, we investigate the behavior of different scalarizing functions and their parameters. We thereby abstract firstly from any specific algorithm and only consider the difficulty of the single scalarized problems in terms of the search ability of a (1+lambda)-EA on biobjective NK-landscapes. Secondly, combining the outcomes of independent single-objective runs allows for more general statements on set-based performance measures. Finally, we investigate the correlation between the opening angle of the scalarizing function's underlying contour lines and the position of the final solution in the objective space. Our analysis is of fundamental nature and sheds more light on the key characteristics of multiobjective scalarizing functions.
Finding a good classifier is a multiobjective optimization problem with different error rates and the costs to be minimized. The receiver operating characteristic is widely used in the machine learning community to analyze the performance of parametric classifiers or sets of Pareto optimal classifiers. In order to directly compare two sets of classifiers the area (or volume) under the convex hull can be used as a scalar indicator for the performance of a set of classifiers in receiver operating characteristic space. Recently, the convex hull based multiobjective genetic programming algorithm was proposed and successfully applied to maximize the convex hull area for binary classification problems. The contribution of this paper is to extend this algorithm for dealing with higher dimensional problem formulations. In particular, we discuss problems where parsimony (or classifier complexity) is stated as a third objective and multi-class classification with three different true classification rates to be maximized. The design of the algorithm proposed in this paper is inspired by indicator-based evolutionary algorithms, where first a performance indicator for a solution set is established and then a selection operator is designed that complies with the performance indicator. In this case, the performance indicator will be the volume under the convex hull. The algorithm is tested and analyzed in a proof of concept study on different benchmarks that are designed for measuring its capability to capture relevant parts of a convex hull. Further benchmark and application studies on email classification and feature selection round up the analysis and assess robustness and usefulness of the new algorithm in real world settings.
Multi-objective optimisation is regarded as one of the most promising ways for dealing with constrained optimisation problems in evolutionary optimisation. This paper presents a theoretical investigation of a multi-objective optimisation evolutionary algorithm for solving the 0-1 knapsack problem. Two initialisation methods are considered in the algorithm: local search initialisation and greedy search initialisation. Then the solution quality of the algorithm is analysed in terms of the approximation ratio.
In the a posteriori approach of multiobjective optimization the Pareto front is approximated by a finite set of solutions in the objective space. The quality of the approximation can be measured by different indicators that take into account the approximation's closeness to the Pareto front and its distribution along the Pareto front. In particular, the averaged Hausdorff indicator prefers an almost uniform distribution. An observed drawback of multiobjective estimation of distribution algorithms (MEDAs) is that - as common for randomized metaheuristics - the final population usually is not uniformly distributed along the Pareto front. Therefore, we propose a postprocessing strategy which consists of applying the averaged Hausdorff indicator to the complete archive of generated solutions after optimization in order to select a uniformly distributed subset of nondominated solutions from the archive. In this paper, we put forward a strategy for extracting the above described subset. The effectiveness of the proposal is contrasted in a series of experiments that involve different MEDAs and filtering techniques.
Path planning is one of the most vital elements of mobile robotics. With a priori knowledge of the environment, global path planning provides a collision-free route through the workspace. The global path plan can be calculated with a variety of informed search algorithms, most notably the A* search method, guaranteed to deliver a complete and optimal solution that minimizes the path cost. Path planning optimization typically looks to minimize the distance traversed from start to goal, yet many mobile robot applications call for additional path planning objectives, presenting a multiobjective optimization (MOO) problem. Past studies have applied genetic algorithms to MOO path planning problems, but these may have the disadvantages of computational complexity and suboptimal solutions. Alternatively, the algorithm in this paper approaches MOO path planning with the use of Pareto fronts, or finding non-dominated solutions. The algorithm presented incorporates Pareto optimality into every step of A* search, thus it is named A*-PO. Results of simulations show A*-PO outperformed several variations of the standard A* algorithm for MOO path planning. A planetary exploration rover case study was added to demonstrate the viability of A*-PO in a real-world application.
Path planning is one of the most vital elements of mobile robotics, providing the agent with a collision-free route through the workspace. The global path plan can be calculated with a variety of informed search algorithms, most notably the A* search method, guaranteed to deliver a complete and optimal solution that minimizes the path cost. D* is widely used for its dynamic replanning capabilities. Path planning optimization typically looks to minimize the distance traversed from start to goal, but many mobile robot applications call for additional path planning objectives, presenting a multiobjective optimization (MOO) problem. Common search algorithms, e.g. A* and D*, are not well suited for MOO problems, yielding suboptimal results. The search algorithm presented in this paper is designed for optimal MOO path planning. The algorithm incorporates Pareto optimality into D*, and is thus named D*-PO. Non-dominated solution paths are guaranteed by calculating the Pareto front at each search step. Simulations were run to model a planetary exploration rover in a Mars environment, with five path costs. The results show the new, Pareto optimal D*-PO outperforms the traditional A* and D* algorithms for MOO path planning.
Random generators or stochastic engines are a key component in the structure of metaheuristic algorithms. This work investigates the effects of non-Gaussian stochastic engines on the performance of metaheuristics when solving a real-world optimization problem. In this work, the bacteria foraging algorithm (BFA) was employed in tandem with four random generators (stochastic engines). The stochastic engines operate using the Weibull distribution, Gamma distribution, Gaussian distribution and a chaotic mechanism. The two non-Gaussian distributions are the Weibull and Gamma distributions. In this work, the approaches developed were implemented on the real-world multi-objective resin bonded sand mould problem. The Pareto frontiers obtained were benchmarked using two metrics; the hyper volume indicator (HVI) and the proposed Average Explorative Rate (AER) metric. Detail discussions from various perspectives on the effects of non-Gaussian random generators in metaheuristics are provided.
Several test function suites are being used for numerical benchmarking of multiobjective optimization algorithms. While they have some desirable properties, like well-understood Pareto sets and Pareto fronts of various shapes, most of the currently used functions possess characteristics that are arguably under-represented in real-world problems. They mainly stem from the easier construction of such functions and result in improbable properties such as separability, optima located exactly at the boundary constraints, and the existence of variables that solely control the distance between a solution and the Pareto front. Here, we propose an alternative way to constructing multiobjective problems-by combining existing single-objective problems from the literature. We describe in particular the bbob-biobj test suite with 55 bi-objective functions in continuous domain, and its extended version with 92 bi-objective functions (bbob-biobj-ext). Both test suites have been implemented in the COCO platform for black-box optimization benchmarking. Finally, we recommend a general procedure for creating test suites for an arbitrary number of objectives. Besides providing the formal function definitions and presenting their (known) properties, this paper also aims at giving the rationale behind our approach in terms of groups of functions with similar properties, objective space normalization, and problem instances. The latter allows us to easily compare the performance of deterministic and stochastic solvers, which is an often overlooked issue in benchmarking.
We introduce COCO, an open source platform for Comparing Continuous Optimizers in a black-box setting. COCO aims at automatizing the tedious and repetitive task of benchmarking numerical optimization algorithms to the greatest possible extent. The platform and the underlying methodology allow to benchmark in the same framework deterministic and stochastic solvers for both single and multiobjective optimization. We present the rationales behind the (decade-long) development of the platform as a general proposition for guidelines towards better benchmarking. We detail underlying fundamental concepts of COCO such as the definition of a problem as a function instance, the underlying idea of instances, the use of target values, and runtime defined by the number of function calls as the central performance measure. Finally, we give a quick overview of the basic code structure and the currently available test suites.
Multi objective (MO) optimization is an emerging field which is increasingly being implemented in many industries globally. In this work, the MO optimization of the extraction process of bioactive compounds from the Gardenia Jasminoides Ellis fruit was solved. Three swarm-based algorithms have been applied in conjunction with normal-boundary intersection (NBI) method to solve this MO problem. The gravitational search algorithm (GSA) and the particle swarm optimization (PSO) technique were implemented in this work. In addition, a novel Hopfield-enhanced particle swarm optimization was developed and applied to the extraction problem. By measuring the levels of dominance, the optimality of the approximate Pareto frontiers produced by all the algorithms were gauged and compared. Besides, by measuring the levels of convergence of the frontier, some understanding regarding the structure of the objective space in terms of its relation to the level of frontier dominance is uncovered. Detail comparative studies were conducted on all the algorithms employed and developed in this work.
Over the last three decades, a large number of evolutionary algorithms have been developed for solving multiobjective optimization problems. However, there lacks an up-to-date and comprehensive software platform for researchers to properly benchmark existing algorithms and for practitioners to apply selected algorithms to solve their real-world problems. The demand of such a common tool becomes even more urgent, when the source code of many proposed algorithms has not been made publicly available. To address these issues, we have developed a MATLAB platform for evolutionary multi-objective optimization in this paper, called PlatEMO, which includes more than 50 multi-objective evolutionary algorithms and more than 100 multi-objective test problems, along with several widely used performance indicators. With a user-friendly graphical user interface, PlatEMO enables users to easily compare several evolutionary algorithms at one time and collect statistical results in Excel or LaTeX files. More importantly, PlatEMO is completely open source, such that users are able to develop new algorithms on the basis of it. This paper introduces the main features of PlatEMO and illustrates how to use it for performing comparative experiments, embedding new algorithms, creating new test problems, and developing performance indicators. Source code of PlatEMO is now available at: http://bimk.ahu.edu.cn/index.php?s=/Index/Software/index.html.
Optimization is becoming a crucial element in industrial applications involving sustainable alternative energy systems. During the design of such systems, the engineer/decision maker would often encounter noise factors (e.g. solar insolation and ambient temperature fluctuations) when their system interacts with the environment. In this chapter, the sizing and design optimization of the solar powered irrigation system was considered. This problem is multivariate, noisy, nonlinear and multiobjective. This design problem was tackled by first using the Fuzzy Type II approach to model the noise factors. Consequently, the Bacterial Foraging Algorithm (BFA) (in the context of a weighted sum framework) was employed to solve this multiobjective fuzzy design problem. This method was then used to construct the approximate Pareto frontier as well as to identify the best solution option in a fuzzy setting. Comprehensive analyses and discussions were performed on the generated numerical results with respect to the implemented solution methods.
One of the most common approaches for multiobjective optimization is to generate a solution set that well approximates the whole Pareto-optimal frontier to facilitate the later decision-making process. However, how to evaluate and compare the quality of different solution sets remains challenging. Existing measures typically require additional problem knowledge and information, such as a reference point or a substituted set of the Pareto-optimal frontier. In this paper, we propose a quality measure, called dominance move (DoM), to compare solution sets generated by multiobjective optimizers. Given two solution sets, DoM measures the minimum sum of move distances for one set to weakly Pareto dominate the other set. DoM can be seen as a natural reflection of the difference between two solutions, capturing all aspects of solution sets' quality, being compliant with Pareto dominance, and does not need any additional problem knowledge and parameters. We present an exact method to calculate the DoM in the biobjective case. We show the necessary condition of constructing the optimal partition for a solution set's minimum move, and accordingly propose an efficient algorithm to recursively calculate the DoM. Finally, DoM is evaluated on several groups of artificial and real test cases as well as by a comparison with two well-established quality measures.
In this paper we introduce a new classification algorithm called Optimization of Distributions Differences (ODD). The algorithm aims to find a transformation from the feature space to a new space where the instances in the same class are as close as possible to one another while the gravity centers of these classes are as far as possible from one another. This aim is formulated as a multiobjective optimization problem that is solved by a hybrid of an evolutionary strategy and the Quasi-Newton method. The choice of the transformation function is flexible and could be any continuous space function. We experiment with a linear and a non-linear transformation in this paper. We show that the algorithm can outperform 6 other state-of-the-art classification methods, namely naive Bayes, support vector machines, linear discriminant analysis, multi-layer perceptrons, decision trees, and k-nearest neighbors, in 12 standard classification datasets. Our results show that the method is less sensitive to the imbalanced number of instances comparing to these methods. We also show that ODD maintains its performance better than other classification methods in these datasets, hence, offers a better generalization ability.
Behavior domination is proposed as a tool for understanding and harnessing the power of evolutionary systems to discover and exploit useful stepping stones. Novelty search has shown promise in overcoming deception by collecting diverse stepping stones, and several algorithms have been proposed that combine novelty with a more traditional fitness measure to refocus search and help novelty search scale to more complex domains. However, combinations of novelty and fitness do not necessarily preserve the stepping stone discovery that novelty search affords. In several existing methods, competition between solutions can lead to an unintended loss of diversity. Behavior domination defines a class of algorithms that avoid this problem, while inheriting theoretical guarantees from multiobjective optimization. Several existing algorithms are shown to be in this class, and a new algorithm is introduced based on fast non-dominated sorting. Experimental results show that this algorithm outperforms existing approaches in domains that contain useful stepping stones, and its advantage is sustained with scale. The conclusion is that behavior domination can help illuminate the complex dynamics of behavior-driven search, and can thus lead to the design of more scalable and robust algorithms.
In this report, we suggest nine test problems for multi-task multi-objective optimization (MTMOO), each of which consists of two multiobjective optimization tasks that need to be solved simultaneously. The relationship between tasks varies between different test problems, which would be helpful to have a comprehensive evaluation of the MO-MFO algorithms. It is expected that the proposed test problems will germinate progress the field of the MTMOO research.
In evolutionary algorithms, a preselection operator aims to select the promising offspring solutions from a candidate offspring set. It is usually based on the estimated or real objective values of the candidate offspring solutions. In a sense, the preselection can be treated as a classification procedure, which classifies the candidate offspring solutions into promising ones and unpromising ones. Following this idea, we propose a classification based preselection (CPS) strategy for evolutionary multiobjective optimization. When applying classification based preselection, an evolutionary algorithm maintains two external populations (training data set) that consist of some selected good and bad solutions found so far; then it trains a classifier based on the training data set in each generation. Finally it uses the classifier to filter the unpromising candidate offspring solutions and choose a promising one from the generated candidate offspring set for each parent solution. In such cases, it is not necessary to estimate or evaluate the objective values of the candidate offspring solutions. The classification based preselection is applied to three state-of-the-art multiobjective evolutionary algorithms (MOEAs) and is empirically studied on two sets of test instances. The experimental results suggest that classification based preselection can successfully improve the performance of these MOEAs.
Recent work has addressed using formulas in linear temporal logic (LTL) as specifications for agents planning in Markov Decision Processes (MDPs). We consider the inverse problem: inferring an LTL specification from demonstrated behavior trajectories in MDPs. We formulate this as a multiobjective optimization problem, and describe state-based ("what actually happened") and action-based ("what the agent expected to happen") objective functions based on a notion of "violation cost". We demonstrate the efficacy of the approach by employing genetic programming to solve this problem in two simple domains.